Set-valued tableaux rule for Lascoux polynomials

TL;DR

This paper introduces a new combinatorial rule for Lascoux polynomials using set-valued tableaux and right keys, unifying formulas for key polynomials and Grassmannian stable Grothendieck polynomials, and constructs a novel crystal structure.

Contribution

It presents a novel combinatorial rule for Lascoux polynomials and constructs a new crystal structure on set-valued tableaux, solving an open problem.

Findings

Established a combinatorial rule for Lascoux polynomials.

Unified formulas for key polynomials and Grothendieck polynomials.

Constructed a new abstract Kashiwara crystal structure.

Abstract

Lascoux polynomials generalize Grassmannian stable Grothendieck polynomials and may be viewed as K-theoretic analogs of key polynomials. The latter two polynomials have combinatorial formulas involving tableaux: Lascoux and Sch\"{u}tzenberger gave a combinatorial formula for key polynomials using right keys; Buch gave a set-valued tableau formula for Grassmannian stable Grothendieck polynomials. We establish a novel combinatorial rule for Lascoux polynomials involving right keys and set-valued tableaux. Our rule recovers the tableaux formulas of key polynomials and Grassmannian stable Grothendieck polynomials. To prove our rule, we construct a new abstract Kashiwara crystal structure on set-valued tableaux. This construction answers an open problem of Monical, Pechenik and Scrimshaw in the context of abstract Kashiwara crystal.